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Restoring Convergence in Heavy-Tailed Risk Models: A Weighted Kolmogorov Approach for Robust Backtesting

Armen Petrosyan

Abstract

Standard risk metrics used in model validation, such as the Kolmogorov-Smirnov distance, fail to converge at practical rates when applied to high-frequency financial data characterized by heavy tails (infinite skewness). This creates a "noise barrier" where valid risk models are rejected due to tail events irrelevant to central tendency accuracy. In this paper, we introduce a Weighted Kolmogorov Metric tailored for financial time series with sub-cubic moments ($\mathbb{E}|X|^{2+δ}<\infty$). By incorporating an exhaustion function $h(x)$ that mechanically downweights extreme tail noise, we prove that we can restore the optimal Gaussian convergence rate of $O(n^{-1/2})$ even for Pareto and Student-t distributions common in Crypto and FX markets. We provide a complete proof using a core/tail truncation scheme and establish the optimal tuning of the weight parameter $q$.

Restoring Convergence in Heavy-Tailed Risk Models: A Weighted Kolmogorov Approach for Robust Backtesting

Abstract

Standard risk metrics used in model validation, such as the Kolmogorov-Smirnov distance, fail to converge at practical rates when applied to high-frequency financial data characterized by heavy tails (infinite skewness). This creates a "noise barrier" where valid risk models are rejected due to tail events irrelevant to central tendency accuracy. In this paper, we introduce a Weighted Kolmogorov Metric tailored for financial time series with sub-cubic moments (). By incorporating an exhaustion function that mechanically downweights extreme tail noise, we prove that we can restore the optimal Gaussian convergence rate of even for Pareto and Student-t distributions common in Crypto and FX markets. We provide a complete proof using a core/tail truncation scheme and establish the optimal tuning of the weight parameter .
Paper Structure (21 sections, 12 theorems, 30 equations, 4 figures)

This paper contains 21 sections, 12 theorems, 30 equations, 4 figures.

Table of Contents

  1. Introduction and Financial Context
  2. Notation.
  3. Weighted Metric for Risk Management
  4. Main Result: Restoring Convergence
  5. Core/tail decomposition and choice of threshold
  6. Extensions and invariance under change of exhaustion
  7. Multivariate extension (Portfolio Optimization)
  8. Numerical Validation and Benchmarks
  9. Scenario 1: Emerging Markets Risk (Student-$t$, $\nu=2.5$)
  10. Scenario 2: Crypto-Asset Risk (Pareto, $\alpha=2.8$)
  11. Interpretation for Risk Managers
  12. Discussion: Operational Implementation and Robustness Framework
  13. Resolving the Core--Tail Paradox via Hybrid Validation
  14. Parameter Selection and an Anti-Gaming Protocol
  15. VaR-Centered Weighting as an Extension (Not a Contradiction)
  16. ...and 6 more sections

Key Result

Proposition 2.4

For any $q>0$ and exhaustion $h$ finite on $\mathbb{R}$, $d_{K,h,q}$ is a metric on the set of CDFs.

Figures (4)

  • Figure 1: Signal vs. Noise. Comparison of convergence rates for Student-$t$ returns ($\nu=2.5$). The Standard Kolmogorov metric (Blue) stagnates at a slow rate ($n^{-0.25}$) due to tail noise. The Weighted Metric (Orange, $q=1.2$) successfully filters outliers and restores the optimal Gaussian convergence rate ($n^{-0.5}$), allowing for faster model validation.
  • Figure 2: Robustness on Crypto-Assets. Convergence of the Weighted Metric for Pareto distributed returns ($\alpha=2.8$). The metric maintains a stable, linear decay in log-log scale (slope $\approx -0.5$), proving its reliability for backtesting VaR models on heavy-tailed asset classes.
  • Figure 3: Hybrid validation: $d_{K,h,q}$ for statistical stability + $\mathcal{T}_{\text{tail}}$ for explicit tail compliance.
  • Figure 4: Stability Certificate via Grid Robustness. Robustness analysis performed on the grid $Q = [0.5, 2.5]$ with a rejection threshold $\epsilon_{core}=0.04$. The Gaussian model (Red) is rejected because its worst-case error ($\approx 0.058$) exceeds the threshold, despite appearing valid for high values of $q$ (illustrating the "gaming" risk). The Student-t model (Green) validates uniformly across the grid (max error $\approx 0.014$), demonstrating structural stability independent of the weighting parameter.

Theorems & Definitions (29)

  • Definition 2.1: Weighted Risk Metric
  • Remark 2.2: Why smooth weighting instead of winsorization/truncation?
  • Remark 2.3: Local Uniform Control for VaR
  • Proposition 2.4: Metric property
  • proof
  • Lemma 3.3: Weight equivalence
  • proof : Proof sketch
  • Remark 3.4: Notation $f\asymp g$ at infinity
  • Theorem 3.5: Global weighted trade-off with explicit dependence
  • Remark 3.6: Constants and dependencies
  • ...and 19 more